202211041114 Solution to 2005-CE-AMATH-II-1

Find $\displaystyle{\int (2x-3)^7\,\mathrm{d}x}$ .

Roughwork.

$\begin{aligned} &\quad\, \int (2x-3)^7\,\mathrm{d}x \\ \dots\enspace & \textrm{let }u=2x-3\textrm{ s.t. }\mathrm{d}x=\frac{\mathrm{d}u}{2} \\ & = \int u^7\bigg( \frac{\mathrm{d}u}{2}\bigg) \\ & = \frac{1}{2} \int u^7\,\mathrm{d}u \\ & = \frac{1}{2}\bigg(\frac{u^8}{8}\bigg) +C\textrm{ for some constant} \\ & = \frac{u^8}{16} + C \\ & = \frac{(2x-3)^8}{16} +C \\ \end{aligned}$

物理後生﹑掣數學之肘.
小子算數﹑指物理之月.

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